I would have thought that the cost of a few of these rotations each year versus the costs of acquiring, crewing and funding the operation of one or more equivalent aircraft (Boeing 737 or 767 tend to be used) would make it a doubtful proposition to have the Air Corps undertake it. The US and UK forces, as you probably know, hire in a good deal of their troop transport capacity from the commercial sector and presumably use their own military transport fleets for more specialised needs.